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High-Speed Gas Bearings for Micro-Turbomachinery

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Multi-Wafer Rotating MEMS Machines

Part of the book series: MEMS Reference Shelf ((MEMSRS))

Abstract

The mechanical design and architecture of high-speed rotating machinery, independent of size or scale, are strongly governed by the rotordynamic behavior of the spool and its bearing arrangement. Large-scale gas turbine engines yield multi-spool shaft constructions where the rolling contact bearings are close to the centerline of the engine supporting the shaft and disk assemblies as shown in Fig. 6.1 on the left.

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Notes

  1. 1.

    Magnetic bearings are not feasible due to Curie point limitations and regular rolling element bearings cannot be manufactured by DRIE.

  2. 2.

    Geometric asymmetries of the rotors are mostly due to local and global variations in etch depth and out-of-roundness effects are negligible. For a more detailed discussion see Chapter 3.

  3. 3.

    The analysis of the hydrostatic thrust bearing dynamics and the tilting degree of freedom are discussed later in Section 6.7

  4. 4.

    At subcritical speeds (below the natural frequency), the rotor is spinning about its geometric center. After crossing the natural frequency and transitioning to supercritical speeds, the rotor is spinning about its center of mass.

  5. 5.

    The effect of unsteady flow on the whirl ratio will be discussed in Section 6.5.2.

  6. 6.

    The bearing number Λ is used to non-dimensionalize the rotational speed Ω and is defined as Λ = 6 μΩ/p 0  · (R/C)2. For an ideal gas the bearing number may also be expressed in variables more familiar to the fluid dynamicist, that is Λ = 6γM 2/Re C · (R/C).

  7. 7.

    The characteristic axial flow-through time is much shorter than the characteristic flow-change time around the circumference such that the reduced frequency becomes \(\beta = {{\Omega L}/{\overline {U}}}\sim 0.05\).

  8. 8.

    Equation (6.33) has the form of the heat diffusion equation, the corresponding heat transfer problem being the sudden increase in temperature of a plate. The analogy is that vorticity spreads like heat.

  9. 9.

    The axial Mach number M z is of order 0.3. For the circumferential flow, numerical studies show that compressibility effects are insignificant for small radial eccentricities, as previously reported in [19].

  10. 10.

    The scaling law for hydrostatic stiffness is K hs/(p 0 C) ∼ (L/D) · (C/R)-2 ·Δp/p 0.

  11. 11.

    It can be shown that if k xy and k yx were of the same sign, the resulting root occurs at even higher frequency and shows a different dynamic behavior that does not constitute whirl instability.

  12. 12.

    The vertical dashed line indicates the simple whirl instability criterion for isotropic bearing configurations, C/R = 2(L/D)2, established in Section 6.6.1.

  13. 13.

    The pressure supply to the thrust bearing supply plenum is held at a constant value, whereas the thrust bearing exhaust is nominally at atmospheric pressure.

  14. 14.

    This corresponds to a normalized dynamic imbalance χ = 1.0.

  15. 15.

    Since the ratio of the axial flow-through time of the leakage flow to the viscous diffusion time is much larger than unity, the flow can be assumed fully developed.

  16. 16.

    The whirl amplitude of the rotor is negligible at very low speeds and the reference waveform is obtained from measurements at very low speed.

  17. 17.

    For devices with single wafer rotors, wafer misalignment does not yield additional imbalance. Wafer misalignment effects become an issue for multi-wafer rotor designs such as the rotor design of a micro-turbocharger [28].

Abbreviations

a :

rotor imbalance

A, AR :

area, area ratio

A 0 :

coefficient

c :

angular damping coefficient

C :

journal bearing clearance, coefficient

C, \(\tilde {C}\) :

damping coefficient

C f :

friction coefficient

d :

orifice diameter

D :

rotor diameter, orifice diameter

D h :

hydraulic diameter

DN :

bearing diameter times rotor speed in mm-rpm

e :

dynamic eccentricity

ê :

unit vector

F :

force, function

G :

Green’s function

h :

thrust-bearing gap or clearance

I :

diametral moment of inertia

I p :

polar moment of inertia

k, K :

stiffness

L :

journal bearing length, orifice length

\(\dot m\) :

mass flow

m :

mass of rotor disk

M :

Mach number, moment

n, N :

harmonic number, number of

p :

pressure,

Δp :

hydrostatic differential pressure across journal bearing

q, Q :

specific flow rate, flow rate

r :

radial coordinate, radial location of orifices

R :

rotor radius, radius of thrust-bearing pad, specific gas constant

Re :

Reynolds number

\(\mathfrak{R}\) :

whirl-ratio Ω W /Ω N

s :

Laplace variable

t :

time

T :

temperature

u, \({\overline{U}}\) :

velocity, mean axial velocity due to hydrostatic flow

v, V :

velocity

\(\mathfrak{W}\) :

whirl number = k v/k p

z :

axial location, parameter

α:

plenum circumferential angle

β :

reduced frequency ωL/\({\overline{U}}\)

β FD :

ratio of axial flow-through time and viscous diffusion time

β x , β y :

Euler angles

δ:

perturbation

ɛ=e/C 0 :

rotor normalized radial eccentricity

φ :

angle

γ :

ratio of specific heats

Λ :

bearing number

χ:

dynamic imbalance

μ :

dynamic viscosity

ν :

kinematic viscosity

ω :

frequency, stagnation pressure loss coefficient

ω inlet :

journal bearing inlet stagnation pressure loss coefficient

Ω:

rotor speed

Ω N :

natural frequency

Ω W :

rotor speed at onset of whirl instability

ψ :

function

ρ, ρ d :

fluid density, rotor disk density

τ w :

wall shear stress

τ d :

characteristic viscous diffusion time

τ f :

flow-through time of axial hydrostatic flow

ζ :

damping ratio, non-dimensional hydrodynamic moment

{}dp :

damping

{}hd :

cross-coupled hydrodynamic

{}hs :

direct-coupled hydrostatic

{}sys :

system

{}p :

hydrodynamic pumping action

{}v :

hydrodynamic viscous effect

{}* :

critical, singular condition

{}a :

ambient

{}o :

nominal, equilibrium condition

{} r :

radial

{} R :

rotor

{} t :

total or stagnation

{} x :

axial

{} θ :

tangential

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Acknowledgments

The work described in this chapter has resulted from contributions of a number of individuals, many of whom are co-authors of the referenced papers. It has been a pleasure working with the students and members of the bearing team of the microengine project. In particular the author would like to mention his gratitude to Dr. L. Liu, Dr. C.J. Teo, Dr. S. Jacobson, and Dr. F. Ehrich. The editorial help by Ms. D. Park is gratefully acknowledged.

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Authors and Affiliations

  1. Gas Turbine Laboratory, Massachusetts Institute of Technology, Bldg 31-265, 77 Massachusetts Avenue, 02139, Cambridge, MA, USA

    Zoltán S. Spakovszky

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Editors and Affiliations

  1. Dept. Electrical Engineering &, Massachusetts Institute of Technology, Massachusetts Ave. 77, Cambridge, 02139-4307, U.S.A.

    Jeffrey Lang

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Spakovszky, Z.S. (2009). High-Speed Gas Bearings for Micro-Turbomachinery. In: Lang, J. (eds) Multi-Wafer Rotating MEMS Machines. MEMS Reference Shelf. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77747-4_6

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  • DOI: https://doi.org/10.1007/978-0-387-77747-4_6

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